Wide-sense stationary processes - Statistical Signal Processing of Complex-Valued Data - page 197

8 Wide-sense stationary processes

The remaining chapters of the topic deal with complex-valued random processes. In
this chapter, we discuss wide-sense stationary (WSS) signals. In Chapter 9 , we look at
nonstationary signals, and in Chapter 10 , we treat cyclostationary signals, which are an
important subclass of nonstationary signals.
Our discussion of WSS signals continues the preliminary exposition given in Sec-
tion 2.6 . WSS processes have shift-invariant second-order statistics, which leads to the
definition of a time-invariant power spectral density (PSD) - an intuitively pleasing idea.
For improper signals, the PSD needs to be complemented by the complementary power
spectral density (C-PSD), which is generally complex-valued. In Section 8.1 , we will see
that WSS processes allow an easy characterization of all possible PSD/C-PSD pairs and
also a spectral representation of the process itself. Section 8.2 discusses widely linear
shift-invariant filtering, with an application to analytic and complex baseband signals.
We also introduce the noncausal widely linear minimum mean-squared error, or Wiener,
filter for estimating a message signal from a noisy measurement. In order to find the
causal approximation of the Wiener filter, we need to adapt existing spectral factoriza-
tion algorithms to the improper case. This is done in Section 8.3 , where we build causal
synthesis, analysis, and Wiener filters for improper WSS vector-valued time series.
Section 8.4 introduces rotary-component and polarization analysis, which are widely
used in a number of research areas, ranging from optics, geophysics, meteorology, and
oceanography to radar. These techniques are usually applied to deterministic signals,
but we present them in a more general stochastic framework. The idea is to represent
a two-dimensional signal in the complex plane as a superposition of ellipses, which
can be analyzed in terms of their shape and orientation. Each ellipse is the sum of a
counterclockwise and a clockwise rotating phasor, called the rotary components. If there
is complete coherence between the rotary components (i.e., they are linearly dependent),
then the signal is completely polarized.
The chapter is concluded with a brief exposition of higher-order statistics of N th-order
stationary signals in Section 8.5 , where we focus on higher-order moment spectra and
the principal domains of analytic signals.

8.1 Spectral representation and power spectral density

Consider a zero-mean wide-sense stationary (WSS) continuous-time complex-valued
random process x ( t )

= u ( t ) + j v ( t ), which is composed from the two real random